Nonintersecting random walks in the neighborhood of a symmetric tacnode

Abstract

Consider a continuous time random walk in $\mathbb{Z}$ with independent and exponentially distributed jumps $\pm 1$. The model in this paper consists in an infinite number of such random walks starting from the complement of $\{-m,-m+1,\dots ,m-1,m\}$ at time -t, returning to the same starting positions at time t, and conditioned not to intersect. This yields a determinantal process, whose gap probabilities are given by the Fredholm determinant of a kernel. Thus this model consists of two groups of random walks, which are contained within two ellipses which, with the choice $m\simeq 2t$ to leading order, just touch: so we have a tacnode. We determine the new limit extended kernel under the scaling $m=\lfloor 2t+\sigma t^{1/3}\rfloor$, where parameter $\sigma$ controls the strength of interaction between the two groups of random walkers.

Published in at http://dx.doi.org/10.1214/11-AOP726 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (12)

• The Pearcey Process
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Nonintersecting Brownian excursions
• Noncolliding Brownian Motion and Determinantal Processes
• Non-colliding Brownian bridges and the asymmetric tacnode process
• Non-colliding Brownian Motions and the extended tacnode process
• From the Pearcey to the Airy process
• Step fluctuations for a faceted crystal

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• The hard-edge tacnode process for Brownian motion
• Limit shape phase transitions: a merger of arctic circles
• Asymptotics of the Tacnode process: a transition between the gap probabilities from the Tacnode to the Airy process
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• Double Interlacing in Random Tiling Models
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• Asymptotic analysis of the two matrix model with a quartic potential
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• Non-intersecting squared Bessel paths with one positive starting and ending point
• Painlevé kernels in Hermitian matrix models